If a and b are two odd positive integers such that a>b, then prove that one of the two numbers (a+b)/2 and (a-b)/2 is odd and other is even.

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## Rahul

## Dhiraj

As we know that an odd positive integer can be written in the form of either (4q + 1) or (4q + 3) where, q ≥ 0.Let a = 4q + 3and b = 4q + 1^{(a+b)}/_{2}=^{(4q+3+4q+1)}/_{2}=^{(8q+4)}/_{2}= 2(2q+1) which is an even integer.^{(a-b)}/_{2}=^{[(4q+3)-(4q+1)]}/_{2}=^{[(4q+3-4q-1)]}/_{2}= 1 which is an odd integer.Hence, one of the two numbers^{(a+b)}/_{2}and^{(a-b)}/_{2}is odd and other is even where a and b are odd positive integers.